Hybridization technique

In this family, most of the processing is done on the interferometric data alone. Indeed, the interferometric data is deconvolved and corrected for the primary beam contribution to obtain

$$I_\ensuremath{\mathrm{clean}}^\ensuremath{\mathrm{int}} = B... ...h{\mathrm{clean}} \star I_\ensuremath{\mathrm{source}} + N',$$ (5.8)

where $B_\ensuremath{\mathrm{clean}}$ is a Gaussian of FWHM equal to the interferometer resolution and $N'$ is some thermal noise corrected for the primary beam contribution. Two main facts are hidden in this formulation: 1) the field-of-view of the observation is obviously limited to the observed portion of the sky and 2) more importantly, the lack of short-spacings has not yet been overcome and a better formulation would be

$$I_\ensuremath{\mathrm{clean}}^\ensuremath{\mathrm{int}} = \... ...m{clean}} \star I_\ensuremath{\mathrm{source}} \right\}} + N'.$$ (5.9)

The simplicity of equation  is thus slightly misleading but we will keep it for the sake of simplicity. The hybridization method consists in combining two images ( $I_\ensuremath{\mathrm{meas}}^\ensuremath{\mathrm{sd}}$ and $I_\ensuremath{\mathrm{clean}}^\ensuremath{\mathrm{int}}$) in the $uv$ plane.

1. Both images are first spatially regridded on the same fine grid.
2. The FFT of those two images are computed, and linearly combined by selecting the low spatial frequencies from FFT( $I_\ensuremath{\mathrm{meas}}^\ensuremath{\mathrm{sd}}$) and the high spatial frequencies from FFT( $I_\ensuremath{\mathrm{clean}}^\ensuremath{\mathrm{int}}$). The transition between low and high spatial frequency
3. The result is FFTed back to the image plane to produce a final,unique image, which takes into account both single-dish and interferometric information.

The method has the following free parameters: the transition radius and the detailed shape of that transition. To avoid discontinuity, the transition shape is chosen to be reasonably smooth. The spatial frequency of transition is generally chosen to the smallest spatial frequency reliably measured by the interferometer (e.g. about 20 m for PdBI).